HullFund8eCh01ProblemSolutions

forward exchange rate. Alternatively the company could buy a call option giving it the right (but not the obligation) to purchase 1 million Canadian dollars at a certain exchange rate in six months. This would provide insurance against a strong Canadian dollar in six months while still allowing the company to benefit from a weak Canadian dollar at that time.

Problem 1.24

A trader buys a call option with a strike price of $30 for $3. Does the trader ever exercise the option and lose money on the trade. Explain.

Yes, the trader will exercise if the asset price is greater than $30, but will cover the cost of the call option only if the price is greater than $33. The trader exercises and loses money if the price is between $30 and $33.

Problem 1.25

A trader sells a put option with a strike price of $40 for $5. What is the trader?s maximum gain and maximum loss? How does your answer change if it is a call option?

The trader’s maximum gain from the put option is $5. The maximum loss is $35,

corresponding to the situation where the option is exercised and the asset price is zero. If the option were a call, the trader’s maximum gain would still be $5, but there would be no bound to the loss as there is in theory no limit to how high the asset price could rise.

Problem 1.26

??Buying a stock and a put option on the stock is a form of insurance.?? Explain this statement.

If the stock price declines below the strike price of the put option, the stock can be sold for the strike price.

Further Questions

Problem 1.27 (Excel file)

Trader A enters into a forward contract to buy an asset for $1000 an ounce in one year. Trader B buys a call option to buy the asset for $1000 in one year. The cost of the option is $100. What is the difference between the positions of the traders? Show the profit as a function of the price of the asset in one year for the two traders.

Trader A makes a profit of ST 1000 and Trader B makes a profit of max(ST 1000, 0) –100 where ST is the price of the asset in one year. Trader A does better if ST is above $900 as indicated in the diagram below

300 200 100 0 700 -1 00 Profit Trader A 900 1100 1300 Asset Price Trader B -2 00 -3 00

Problem 1.28

On June 25, 2012, as indicated in Table 1.2, the spot offer price of Google stock is $561.51 and the offer price of a call option with a strike price of $560 and a maturity date of

September is $30.70. A trader is considering two alternatives: buy 100 shares of the stock and buy 100 September call options. For each alternative, what is (a) the upfront cost, (b) the total gain if the stock price in September is $620, and (c) the total loss if the stock price in September is $500. Assume that the option is not exercised before September and if stock is purchased it is sold in September.

a) The upfront cost for the share alternative is $56,151. The upfront cost for the option alternative is $3,070.

b) The gain from the stock alternative is $62,000?$56,151=$5,849. The total gain from the option alternative is ($620-$560)×100?$3,070=$2,930.

c) The loss from the stock alternative is $56,151?$50,000=$6,151. The loss from the option alternative is $3,070.

Problem 1.29

What is arbitrage? Explain the arbitrage opportunity when the price of a dually listed

mining company stock is $50 on the New York Stock Exchange and $52 CAD on the Toronto Stock Exchange. Assume that the exchange rate is such that 1 USD equals 1.01 CAD. Explain what is likely to happen to prices as traders take advantage of this opportunity.

Arbitrage involves carrying out two or more different trades to lock in a profit. In this case, traders can buy shares on the NYSE and sell them on the TSX to lock in a USD profit of 52/1.01?50=1.485 per share. As they do this the NYSE price will rise and the TSX price will fall so that the arbitrage opportunity disappears

Problem 1.30

In March, a US investor instructs a broker to sell one July put option contract on a stock. The stock price is $42 and the strike price is $40. The option price is $3. Explain what the

investor has agreed to. Under what circumstances will the trade prove to be profitable? What are the risks?

The investor has agreed to buy 100 shares of the stock for $40 in July (or earlier) if the party on the other side of the transaction chooses to sell. The trade will prove profitable if the option is not exercised or if the stock price is above $37 at the time of exercise. The risk to the investor is that the stock price plunges to a low level. For example, if the stock price drops to $1 by July (unlikely but possible), the investor loses $3,600. This is because the put options are exercised and $40 is paid for 100 shares when the value per share is $1. This leads to a loss of $3,900 which is offset by the premium of $300 received for the options.

Problem 1.31

A US company knows it will have to pay 3 million euros in three months. The current

exchange rate is 1.4500 dollars per euro. Discuss how forward and options contracts can be used by the company to hedge its exposure.

The company could enter into a forward contract obligating it to buy 3 million euros in three months for a fixed price (the forward price). The forward price will be close to but not exactly the same as the current spot price of 1.4500. An alternative would be to buy a call option giving the company the right but not the obligation to buy 3 million euros for a a

particular exchange rate (the strike price) in three months. The use of a forward contract locks in, at no cost, the exchange rate that will apply in three months. The use of a call option provides, at a cost, insurance against the exchange rate being higher than the strike price.

Problem 1.32 (Excel file)

A stock price is $29. An investor buys one call option contract on the stock with a strike price of $30 and sells a call option contract on the stock with a strike price of $32.50. The market prices of the options are $2.75 and $1.50, respectively. The options have the same maturity date. Describe the investor's position.

This is known as a bull spread and will be discussed in Chapter 11. The profit is shown in the diagram below

Problem 1.33

The price of gold is currently $1,800 per ounce. Forward contracts are available to buy or sell gold at $2,000 per ounce for delivery in one year. An arbitrageur can borrow money at 5% per annum. What should the arbitrageur do? Assume that the cost of storing gold is zero and that gold provides no income.

The arbitrageur should borrow money to buy a certain number of ounces of gold today and short forward contracts on the same number of ounces of gold for delivery in one year. This means that gold is purchased for $1800 per ounce and sold for $2000 per ounce. The cost of the borrowing is $90 per ounce. A riskless profit of $110 per ounce is generated.

Problem 1.34.

Discuss how foreign currency options can be used for hedging in the situation described in Example 1.1 so that (a) ImportCo is guaranteed that its exchange rate will be less than 1.5800, and (b) ExportCo is guaranteed that its exchange rate will be at least 1.5400.

ImportCo can buy call options on $10,000,000 with a strike price of 1.5800. This will ensure that it never pays more than $15,800,000 for the sterling it requires. ExportCo can buy put options on $30,000,000 with a strike price of 1.5400. This will ensure that the price received for the sterling will be above1.54?30,000,000?$46,200,00.

Problem 1.35.

The current price of a stock is $94, and three-month call options with a strike price of $95 currently sell for $4.70. An investor who feels that the price of the stock will increase is trying to decide between buying 100 shares and buying 2,000 call options (20 contracts). Both strategies involve an investment of $9,400. What advice would you give? How high does the stock price have to rise for the option strategy to be more profitable?

The investment in call options entails higher risks but can lead to higher returns. If the stock price stays at $94, an investor who buys call options loses $9,400 whereas an investor who buys shares neither gains nor loses anything. If the stock price rises to $120, the investor who buys call options gains 2000?(120?95)?9400?$40?600 An investor who buys shares gains 100?(120?94)?$2?600

The strategies are equally profitable if the stock price rises to a level, S, where 100?(S?94)?2000(S?95)?9400 or

S?100

The option strategy is therefore more profitable if the stock price rises above $100.

Problem 1.36.

On June 25, 2012, an investor owns 100 Google shares. As indicated in Table 1.3, the bid share price is $561.32 and a December put option with a strike price $520 costs $26.10. The

investor is comparing two alternatives to limit downside risk. The first involves buying one Decmber put option contract with a strike price of $520. The second involves instructing a broker to sell the 100 shares as soon as Google?s price reaches $520. Discuss the advantages and disadvantages of the two strategies.

The second alternative involves what is known as a stop or stop-loss order. It costs nothing and ensures that $52,000, or close to $52,000, is realized for the holding in the event the

stock price ever falls to $520. The put option costs $2,610 and guarantees that the holding can be sold for $52,000 any time up to December. If the stock price falls marginally below $520 and then rises the option will not be exercised, but the stop-loss order will lead to the holding being liquidated. There are some circumstances where the put option alternative leads to a better outcome and some circumstances where the stop-loss order leads to a better outcome. If the stock price ends up below $520, the stop-loss order alternative leads to a better outcome because the cost of the option is avoided. If the stock price falls to $480 in

November and then rises to $580 by December, the put option alternative leads to a better outcome. The investor is paying $2,610 for the chance to benefit from this second type of outcome.

Problem 1.37.

A trader buys a European call option and sells a European put option. The options have the same underlying asset, strike price and maturity. Describe the trader?s position. Under what circumstances does the price of the call equal the price of the put?

The trader has a long European call option with strike price K and a short European put option with strike price K. Suppose the price of the underlying asset at the maturity of the option is ST. If ST?K, the call option is exercised by the investor and the put option expires worthless. The payoff from the portfolio isST?K. If ST?K, the call option expires worthless and the put option is exercised against the investor. The cost to the investor is K?ST. Alternatively we can say that the payoff to the investor is ST?K (a negative

amount). In all cases, the payoff isST?K, the same as the payoff from the forward contract. The trader’s position is equivalent to a forward contract with delivery priceK.

Suppose thatFis the forward price. If K?F, the forward contract that is created has zero value. Because the forward contract is equivalent to a long call and a short put, this shows that the price of a call equals the price of a put when the strike price isF.